data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Four of the following test scores are Henry's and the other four are Julia's: 88, 90, 92, 94, 95, 97, 98, 99. Henry's mean score is 94. What is Julia's mean score? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "94.25",
"style": "rule"
} | {
"index": "38600",
"split": "train"
} |
math | [
{
"content": "Each face of a rectangular prism is painted with a single narrow stripe from one vertex to the diagonally opposite vertex. The choice of the vertex pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the rectangular prism?... | {
"ground_truth": "\\frac{3}{16}",
"style": "rule"
} | {
"index": "38601",
"split": "train"
} |
math | [
{
"content": "If you roll four standard, fair six-sided dice, the top faces of the dice can show just one value (for example, $3333$ ), two values (for example, $2666$ ), three values (for example, $5215$ ), or four values (for example, $4236$ ). The mean number of values that show is $\\frac{m}{n}$ , whe... | {
"ground_truth": "887",
"style": "rule"
} | {
"index": "38602",
"split": "train"
} |
math | [
{
"content": "Given a geometric sequence $\\{a_n\\}$ satisfies $a_1=3$, and $a_1+a_3+a_5=21$. Calculate the value of $a_3+a_5+a_7$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "42",
"style": "rule"
} | {
"index": "38603",
"split": "train"
} |
math | [
{
"content": "Let $n \\ge 2$ be an integer. Alex writes the numbers $1, 2, ..., n$ in some order on a circle such that any two neighbours are coprime. Then, for any two numbers that are not comprime, Alex draws a line segment between them. For each such segment $s$ we denote by $d_s$ the difference of t... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "38604",
"split": "train"
} |
math | [
{
"content": "In an $n \\times n$ matrix $\\begin{pmatrix} 1 & 2 & 3 & … & n-2 & n-1 & n \\\\ 2 & 3 & 4 & … & n-1 & n & 1 \\\\ 3 & 4 & 5 & … & n & 1 & 2 \\\\ … & … & … & … & … & … & … \\\\ n & 1 & 2 & … & n-3 & n-2 & n-1\\\\end{pmatrix}$, if the number at the $i$-th row and $j$-th column is denoted as $a_{ij}(i... | {
"ground_truth": "88",
"style": "rule"
} | {
"index": "38605",
"split": "train"
} |
math | [
{
"content": "A bag has 3 red marbles and 5 white marbles. Two marbles are drawn from the bag and not replaced. What is the probability that the first marble is red and the second marble is white? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{15}{56}",
"style": "rule"
} | {
"index": "38606",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $\\sin (A-B)+ \\sin C= \\sqrt {2}\\sin A$.\n(I) Find the value of angle $B$;\n(II) If $b=2$, find the maximum value of $a^{2}+c^{2}$, and find the values of angles $A$ and $C$ when the maximum value ... | {
"ground_truth": "8+4 \\sqrt {2}",
"style": "rule"
} | {
"index": "38607",
"split": "train"
} |
math | [
{
"content": "What is the minimum number of digits to the right of the decimal point needed to express the fraction $\\frac{987654321}{2^{30} \\cdot 5^3}$ as a decimal? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "38608",
"split": "train"
} |
math | [
{
"content": "Given that $21^{-1} \\equiv 15 \\pmod{61}$, find $40^{-1} \\pmod{61}$, as a residue modulo 61. (Provide a number between 0 and 60, inclusive.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "46",
"style": "rule"
} | {
"index": "38609",
"split": "train"
} |
math | [
{
"content": "Evaluate $97 \\times 97$ in your head. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9409",
"style": "rule"
} | {
"index": "38610",
"split": "train"
} |
math | [
{
"content": "You are given that $x$ is directly proportional to $y^3$, and $y$ is inversely proportional to $\\sqrt{z}$. If the value of $x$ is 3 when $z$ is $12$, what is the value of $x$ when $z$ is equal to $75$? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\frac{24}{125}",
"style": "rule"
} | {
"index": "38611",
"split": "train"
} |
math | [
{
"content": "Given the lengths of the three sides of $\\triangle ABC$ are $AB=7$, $BC=5$, and $CA=6$, the value of $\\overrightarrow {AB}\\cdot \\overrightarrow {BC}$ is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-19",
"style": "rule"
} | {
"index": "38612",
"split": "train"
} |
math | [
{
"content": "Given that $\\binom{18}{8}=31824$, $\\binom{18}{9}=48620$, and $\\binom{18}{10}=43758$, calculate $\\binom{20}{10}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "172822",
"style": "rule"
} | {
"index": "38613",
"split": "train"
} |
math | [
{
"content": "Let $p$ be a real number and $c \\neq 0$ an integer such that $c-0.1<x^{p}\\left(\\frac{1-(1+x)^{10}}{1+(1+x)^{10}}\\right)<c+0.1$ for all (positive) real numbers $x$ with $0<x<10^{-100}$. Find the ordered pair $(p, c)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(-1, -5)",
"style": "rule"
} | {
"index": "38614",
"split": "train"
} |
math | [
{
"content": "Vasya is creating a 4-digit password for a combination lock. He dislikes the digit 2, so he does not use it. Additionally, he dislikes when two identical digits stand next to each other. Furthermore, he wants the first digit to match the last. How many possible combinations must be tried to guaran... | {
"ground_truth": "504",
"style": "rule"
} | {
"index": "38615",
"split": "train"
} |
math | [
{
"content": "Carefully observe the following three rows of related numbers:<br/>First row: $-2$, $4$, $-8$, $16$, $-32$, $\\ldots$;<br/>Second row: $0$, $6$, $-6$, $18$, $-30$, $\\ldots$;<br/>Third row: $-1$, $2$, $-4$, $8$, $-16$, $\\ldots$;<br/>Answer the following questions:<br/>$(1)$ The $6$th number in th... | {
"ground_truth": "256",
"style": "rule"
} | {
"index": "38616",
"split": "train"
} |
math | [
{
"content": "The diagram shows the ellipse whose equation is \\(x^{2}+y^{2}-xy+x-4y=12\\). The curve cuts the \\(y\\)-axis at points \\(A\\) and \\(C\\) and cuts the \\(x\\)-axis at points \\(B\\) and \\(D\\). What is the area of the inscribed quadrilateral \\(ABCD\\)? Please output the final answer within \\b... | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "38617",
"split": "train"
} |
math | [
{
"content": "Given $$\\overrightarrow {a} = (x-1, y)$$, $$\\overrightarrow {b} = (x+1, y)$$, and $$|\\overrightarrow {a}| + |\\overrightarrow {b}| = 4$$ \n(1) Find the equation of the trajectory C of point M(x, y). \n(2) Let P be a moving point on curve C, and F<sub>1</sub>(-1, 0), F<sub>2</sub>(1, 0), find ... | {
"ground_truth": "\\frac {2 \\sqrt {21}}{7}",
"style": "rule"
} | {
"index": "38618",
"split": "train"
} |
math | [
{
"content": "Define $g$ by $g(x)=5x-4$. If $g(x)=f^{-1}(x)-3$ and $f^{-1}(x)$ is the inverse of the function $f(x)=ax+b$, find $5a+5b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "38619",
"split": "train"
} |
math | [
{
"content": "In acute triangle $\\triangle ABC$, $a$, $b$, $c$ are the lengths of the sides opposite to angles $A$, $B$, $C$ respectively, and $4a\\sin B = \\sqrt{7}b$. \n$(1)$ If $a = 6$ and $b+c = 8$, find the area of $\\triangle ABC$. \n$(2)$ Find the value of $\\sin (2A+\\frac{2\\pi}{3})$. Please output ... | {
"ground_truth": "\\frac{\\sqrt{3}-3\\sqrt{7}}{16}",
"style": "rule"
} | {
"index": "38620",
"split": "train"
} |
math | [
{
"content": "Triangles $ABC$ and $ADF$ have areas $4014$ and $14007,$ respectively, with $B=(0,0), C=(447,0), D=(1360,760),$ and $F=(1378,778).$ What is the sum of all possible $x$-coordinates of $A$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2400",
"style": "rule"
} | {
"index": "38621",
"split": "train"
} |
math | [
{
"content": "There are 2010 boxes labeled $B_1, B_2, \\dots, B_{2010}$, and $2010n$ balls have been distributed\namong them, for some positive integer $n$. You may redistribute the balls by a sequence of moves,\neach of which consists of choosing an $i$ and moving \\emph{exactly} $i$ balls from box $B_i$ into ... | {
"ground_truth": "n \\geq 1005",
"style": "rule"
} | {
"index": "38622",
"split": "train"
} |
math | [
{
"content": "$\\triangle ABC\\sim\\triangle DBE$, $BC=20\\text{ cm}.$ How many centimeters long is $DE$? Express your answer as a decimal to the nearest tenth. [asy]\ndraw((0,0)--(20,0)--(20,12)--cycle);\ndraw((13,0)--(13,7.8));\nlabel(\"$B$\",(0,0),SW);\nlabel(\"$E$\",(13,0),S);\nlabel(\"$D$\",(13,7.8),NW);\n... | {
"ground_truth": "7.8",
"style": "rule"
} | {
"index": "38623",
"split": "train"
} |
math | [
{
"content": "$(1+x^2)(1-x^3)$ equals Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1+x^2-x^3-x^5",
"style": "rule"
} | {
"index": "38624",
"split": "train"
} |
math | [
{
"content": "Marty wants to paint a box. He can choose to use either blue, green, yellow, or black paint. Also, he can style the paint by painting with a brush, a roller, or a sponge. How many different combinations of color and painting method can Marty choose? Please output the final answer within \\boxed... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "38625",
"split": "train"
} |
math | [
{
"content": "What is the area enclosed by the region defined by the equation $x^2+y^2+10x+24y=0$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "169\\pi",
"style": "rule"
} | {
"index": "38626",
"split": "train"
} |
math | [
{
"content": "Given a triangle \\(PQR\\). Point \\(T\\) is the center of the inscribed circle.\n\nThe rays \\(PT\\) and \\(QT\\) intersect side \\(PQ\\) at points \\(E\\) and \\(F\\) respectively. It is known that the areas of triangles \\(PQR\\) and \\(TFE\\) are equal. What part of side \\(PQ\\) constitutes f... | {
"ground_truth": "\\frac{3 - \\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "38627",
"split": "train"
} |
math | [
{
"content": "Find the smallest prime number $p$ that cannot be represented in the form $|3^{a} - 2^{b}|$, where $a$ and $b$ are non-negative integers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "41",
"style": "rule"
} | {
"index": "38628",
"split": "train"
} |
math | [
{
"content": "Consider a 4x4 grid with points that are equally spaced horizontally and vertically, where the distance between two neighboring points is 1 unit. Two triangles are formed: Triangle A connects points at (0,0), (3,2), and (2,3), while Triangle B connects points at (0,3), (3,3), and (3,0). What is th... | {
"ground_truth": "0.5",
"style": "rule"
} | {
"index": "38629",
"split": "train"
} |
math | [
{
"content": "Let $a+a_{1}(x+2)+a_{2}(x+2)^{2}+\\ldots+a_{12}(x+12)^{12}=(x^{2}-2x-2)^{6}$, where $a_{i}$ are constants. Find the value of $2a_{2}+6a_{3}+12a_{4}+20a_{5}+\\ldots+132a_{12}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "492",
"style": "rule"
} | {
"index": "38630",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbf{R}$ be the matrix for rotating about the origin counter-clockwise by an angle of $58^\\circ.$ Find $\\det \\mathbf{R}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "38631",
"split": "train"
} |
math | [
{
"content": "The teacher fills some numbers into the circles in the diagram (each circle can and must contain only one number). The sum of the three numbers in each of the left and right closed loops is 30, and the sum of the four numbers in each of the top and bottom closed loops is 40. If the number in circl... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "38632",
"split": "train"
} |
math | [
{
"content": "How many distinct sets are there such that each set contains only non-negative powers of $2$ or $3$ and sum of its elements is $2014$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "38633",
"split": "train"
} |
math | [
{
"content": "A positive integer \\( n \\) is said to be good if \\( 3n \\) is a re-ordering of the digits of \\( n \\) when they are expressed in decimal notation. Find a four-digit good integer which is divisible by 11. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2475",
"style": "rule"
} | {
"index": "38634",
"split": "train"
} |
math | [
{
"content": "What is the greatest whole number that must be a divisor of the product of any three consecutive positive integers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "38635",
"split": "train"
} |
math | [
{
"content": "In the diagram, triangles $ABC$ and $CDE$ are isosceles. The perimeter of $\\triangle CDE$ is $22,$ the perimeter of $\\triangle ABC$ is $24,$ and the length of $CE$ is $9.$ What is the length of $AB?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "38636",
"split": "train"
} |
math | [
{
"content": "If $f(x) = \\frac{1 + x}{1 - 3x}, f_1(x) = f(f(x)), f_2(x) = f(f_1(x)),$ and in general $f_n(x) = f(f_{n-1}(x)),$ then $f_{2023}(2)=$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{7}",
"style": "rule"
} | {
"index": "38637",
"split": "train"
} |
math | [
{
"content": "A reservoir is equipped with an inlet tap A and outlet taps B and C. It is known that opening taps A and B together for 8 hours or opening taps A, B, and C together for a whole day can fill an empty reservoir, and opening taps B and C together for 8 hours can empty a full reservoir. Determine how ... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "38638",
"split": "train"
} |
math | [
{
"content": "In the figure, $\\angle EAB$ and $\\angle ABC$ are right angles, $AB=4,$ $BC=6,$ $AE=8,$ and $\\overline{AC}$ and $\\overline{BE}$ intersect at $D.$ What is the difference between the areas of $\\triangle ADE$ and $\\triangle BDC?$ [asy]\npair A,B,C,D,I;\nI=(0,8);\nA=(0,0);\nB=(4,0);\nC=(4,6);\nD=... | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "38639",
"split": "train"
} |
math | [
{
"content": "Find the integer that is closest to $1000\\sum_{n=3}^{10000}\\frac1{n^2-4}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "521",
"style": "rule"
} | {
"index": "38640",
"split": "train"
} |
math | [
{
"content": "The least positive angle $\\alpha$ for which $$ \\left(\\frac34-\\sin^2(\\alpha)\\right)\\left(\\frac34-\\sin^2(3\\alpha)\\right)\\left(\\frac34-\\sin^2(3^2\\alpha)\\right)\\left(\\frac34-\\sin^2(3^3\\alpha)\\right)=\\frac1{256} $$ has a degree measure of $\\tfrac{m}{n}$ , where $m$ and ... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "38641",
"split": "train"
} |
math | [
{
"content": "In triangle $ΔABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $a=\\sqrt{3}$, $b=\\sqrt{2}$, and $A=\\frac{\\pi}{3}$, find the value of $B=$ _______; and the area of $ΔABC=S_{ΔABC}=$ _______. Please output the final answer within \\boxe... | {
"ground_truth": "\\frac{3+ \\sqrt{3}}{4}",
"style": "rule"
} | {
"index": "38642",
"split": "train"
} |
math | [
{
"content": "Given in $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$, and $\\sin A=\\sqrt{3}(1-\\cos A)$.\n\n$(1)$ Find $A$;\n\n$(2)$ If $a=7$ and $\\sin B+\\sin C=\\frac{13\\sqrt{3}}{14}$, find the area of $\\triangle ABC$. Please output the final answer wi... | {
"ground_truth": "10 \\sqrt{3}",
"style": "rule"
} | {
"index": "38643",
"split": "train"
} |
math | [
{
"content": "Given an geometric sequence \\\\(\\{a_n\\}\\) with a common ratio less than \\\\(1\\\\), the sum of the first \\\\(n\\\\) terms is \\\\(S_n\\\\), and \\\\(a_1 = \\frac{1}{2}\\\\), \\\\(7a_2 = 2S_3\\\\).\n\n\\\\((1)\\\\) Find the general formula for the sequence \\\\(\\{a_n\\}\\).\n\n\\\\((2)\\\\) ... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "38644",
"split": "train"
} |
math | [
{
"content": "Let \\( x \\) be a real number satisfying \\( x^{2} - \\sqrt{6} x + 1 = 0 \\). Find the numerical value of \\( \\left| x^{4} - \\frac{1}{x^{4}} \\right|. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\sqrt{2}",
"style": "rule"
} | {
"index": "38645",
"split": "train"
} |
math | [
{
"content": "The sum\\[\\sum_{x=2}^{44} 2\\sin{x}\\sin{1}[1 + \\sec (x-1) \\sec (x+1)]\\]can be written in the form $\\sum_{n=1}^{4} (-1)^n \\frac{\\Phi(\\theta_n)}{\\Psi(\\theta_n)}$, where $\\Phi,\\, \\Psi$ are trigonometric functions and $\\theta_1,\\, \\theta_2, \\, \\theta_3, \\, \\theta_4$ are degrees $\... | {
"ground_truth": "92",
"style": "rule"
} | {
"index": "38646",
"split": "train"
} |
math | [
{
"content": "Fill in numbers in the boxes below so that the sum of the entries in each three consecutive boxes is $2005$ . What is the number that goes into the leftmost box?\n\n[asy]\nsize(300);\nlabel(\"999\",(2.5,.5));\nlabel(\"888\",(7.5,.5));\n\ndraw((0,0)--(9,0));\ndraw((0,1)--(9,1));\nfor (int i=0; i<=... | {
"ground_truth": "118",
"style": "rule"
} | {
"index": "38647",
"split": "train"
} |
math | [
{
"content": "Let point $P$ be the intersection point in the first quadrant of the hyperbola $\\frac{x^{2}}{a^{2}}- \\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ and the circle $x^{2}+y^{2}=a^{2}+b^{2}$. Let $F\\_1$ and $F\\_2$ be the left and right foci of the hyperbola, respectively, and $|PF\\_1| = 2|PF\\_2|$. Find... | {
"ground_truth": "\\sqrt{5}",
"style": "rule"
} | {
"index": "38648",
"split": "train"
} |
math | [
{
"content": "Construct a circle with center \\( S \\) and radius \\( 3 \\text{ cm} \\). Construct two mutually perpendicular diameters \\( AC \\) and \\( BD \\) of this circle. Construct isosceles triangles \\( ABK, BCL, CDM, DAN \\) such that:\n\n- The base of each triangle is a side of the quadrilateral \\( ... | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "38649",
"split": "train"
} |
math | [
{
"content": "In a bag, there are a total of 12 balls that are identical in shape and mass, including red, black, yellow, and green balls. When drawing a ball at random, the probability of getting a red ball is $\\dfrac{1}{3}$, the probability of getting a black or yellow ball is $\\dfrac{5}{12}$, and the proba... | {
"ground_truth": "\\dfrac{5}{12}",
"style": "rule"
} | {
"index": "38650",
"split": "train"
} |
math | [
{
"content": "Given that the area of obtuse triangle $\\triangle ABC$ is $2 \\sqrt {3}$, $AB=2$, $BC=4$, find the radius of the circumcircle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac {2 \\sqrt {21}}{3}",
"style": "rule"
} | {
"index": "38651",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$ with sides $AB=9$, $AC=3$, and $BC=8$, the angle bisector $AK$ is drawn. A point $M$ is marked on side $AC$ such that $AM : CM=3 : 1$. Point $N$ is the intersection point of $AK$ and $BM$. Find $KN$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{15}}{5}",
"style": "rule"
} | {
"index": "38652",
"split": "train"
} |
math | [
{
"content": "How many five-digit positive integers have the product of their digits equal to 1000? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "38653",
"split": "train"
} |
math | [
{
"content": "Each cell of an $m\\times n$ board is filled with some nonnegative integer. Two numbers in the filling are said to be adjacent if their cells share a common side. (Note that two numbers in cells that share only a corner are not adjacent). The filling is called a garden if it satisfies the followin... | {
"ground_truth": "\\boxed{2^{mn} - 1}",
"style": "rule"
} | {
"index": "38654",
"split": "train"
} |
math | [
{
"content": "For $1 \\leq i \\leq 215$ let $a_i = \\dfrac{1}{2^{i}}$ and $a_{216} = \\dfrac{1}{2^{215}}$. Let $x_1, x_2, ..., x_{216}$ be positive real numbers such that $\\sum_{i=1}^{216} x_i=1$ and $\\sum_{1 \\leq i < j \\leq 216} x_ix_j = \\dfrac{107}{215} + \\sum_{i=1}^{216} \\dfrac{a_i x_i^{2}}{2(1-a_i)}$... | {
"ground_truth": "863",
"style": "rule"
} | {
"index": "38655",
"split": "train"
} |
math | [
{
"content": "A particular novel contains $40,\\!000$ words. If the author completed the novel in $80$ hours, on average how many words per hour did she write? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "500",
"style": "rule"
} | {
"index": "38656",
"split": "train"
} |
math | [
{
"content": "Given \\( S = [\\sqrt{1}] + [\\sqrt{2}] + \\cdots + [\\sqrt{1988}] \\), find \\( [\\sqrt{S}] \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "241",
"style": "rule"
} | {
"index": "38657",
"split": "train"
} |
math | [
{
"content": "Sequence $(a_n)$ is defined as $a_{n+1}-2a_n+a_{n-1}=7$ for every $n\\geq 2$ , where $a_1 = 1, a_2=5$ . What is $a_{17}$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "905",
"style": "rule"
} | {
"index": "38658",
"split": "train"
} |
math | [
{
"content": "A circle is inscribed in a regular hexagon. A smaller hexagon has two non-adjacent sides coinciding with the sides of the larger hexagon and the remaining vertices touching the circle. What percentage of the area of the larger hexagon is the area of the smaller hexagon? Assume the side of the larg... | {
"ground_truth": "25\\%",
"style": "rule"
} | {
"index": "38659",
"split": "train"
} |
math | [
{
"content": "Several students are seated at a large circular table. They pass around a bag containing $100$ pieces of candy. Each person receives the bag, takes one piece of candy and then passes the bag to the next person. If Chris takes the first and last piece of candy, then the number of students at the... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "38660",
"split": "train"
} |
math | [
{
"content": "The average of the numbers $1, 2, 3,\\dots, 50,$ and $x$ is $51x$. What is $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{51}{104}",
"style": "rule"
} | {
"index": "38661",
"split": "train"
} |
math | [
{
"content": "Let \\( S \\) be the set comprising all sides and diagonals of a regular hexagon. A pair of elements from \\( S \\) are selected randomly without replacement. What is the probability that the two chosen segments have the same length? Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "\\frac{11}{35}",
"style": "rule"
} | {
"index": "38662",
"split": "train"
} |
math | [
{
"content": "Given the set $A={3,3^{2},3^{3},…,3^{n}}$ $(n\\geqslant 3)$, choose three different numbers from it and arrange them in a certain order to form a geometric sequence. Denote the number of geometric sequences that satisfy this condition as $f(n)$.\n(I) Find $f(5)=$ _______ ;\n(II) If $f(n)=220$, fin... | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "38663",
"split": "train"
} |
math | [
{
"content": "Compute the following expressions:\n\n(1) $2 \\sqrt{12} -6 \\sqrt{ \\frac{1}{3}} + \\sqrt{48}$\n\n(2) $(\\sqrt{3}-\\pi)^{0}-\\frac{\\sqrt{20}-\\sqrt{15}}{\\sqrt{5}}+(-1)^{2017}$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3} - 2",
"style": "rule"
} | {
"index": "38664",
"split": "train"
} |
math | [
{
"content": "Line segment $\\overline{AB}$ has perpendicular bisector $\\overline{CD}$ , where $C$ is the midpoint of $\\overline{AB}$ . The segments have lengths $AB = 72$ and $CD = 60$ . Let $R$ be the set of points $P$ that are midpoints of line segments $\\overline{XY}$ , where $X$ lies on... | {
"ground_truth": "1080",
"style": "rule"
} | {
"index": "38665",
"split": "train"
} |
math | [
{
"content": "Express the following sum as a common fraction: $$\\frac{1}{1\\cdot 2} + \\frac{1}{2\\cdot 3} + \\frac{1}{3\\cdot 4} + \\frac{1}{4\\cdot 5} + \\dots + \\frac{1}{9\\cdot 10}$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9}{10}",
"style": "rule"
} | {
"index": "38666",
"split": "train"
} |
math | [
{
"content": "Given that $α$ is an angle in the fourth quadrant and $\\cos α= \\frac {3}{5}$.\n(1) Find the value of $\\tan α$;\n(2) Find the value of $\\frac {\\sin ( \\frac {3}{2}π-α)+2\\cos (α+ \\frac {π}{2})}{\\sin (α-π)-3\\cos (2π-α)}$. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "38667",
"split": "train"
} |
math | [
{
"content": "At the end of which year did Steve have more money than Wayne for the first time? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2004",
"style": "rule"
} | {
"index": "38668",
"split": "train"
} |
math | [
{
"content": "The line passing through the point $(0,-2)$ intersects the parabola $y^{2}=16x$ at two points $A(x_1,y_1)$ and $B(x_2,y_2)$, with $y_1^2-y_2^2=1$. Calculate the area of the triangle $\\triangle OAB$, where $O$ is the origin. Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "\\frac{1}{16}",
"style": "rule"
} | {
"index": "38669",
"split": "train"
} |
math | [
{
"content": "If I choose four cards from a standard $52$-card deck, with replacement, what is the probability that I will end up with one card from each suit? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{32}",
"style": "rule"
} | {
"index": "38670",
"split": "train"
} |
math | [
{
"content": "How many natural numbers greater than 10 but less than 100 are relatively prime to 21? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "53",
"style": "rule"
} | {
"index": "38671",
"split": "train"
} |
math | [
{
"content": "Given the tower function $T(n)$ defined by $T(1) = 3$ and $T(n + 1) = 3^{T(n)}$ for $n \\geq 1$, calculate the largest integer $k$ for which $\\underbrace{\\log_3\\log_3\\log_3\\ldots\\log_3B}_{k\\text{ times}}$ is defined, where $B = (T(2005))^A$ and $A = (T(2005))^{T(2005)}$. Please output the f... | {
"ground_truth": "2005",
"style": "rule"
} | {
"index": "38672",
"split": "train"
} |
math | [
{
"content": "Given a set $S$ consisting of two undefined elements \"pib\" and \"maa\", and the four postulates: $P_1$: Every pib is a collection of maas, $P_2$: Any two distinct pibs have one and only one maa in common, $P_3$: Every maa belongs to two and only two pibs, $P_4$: There are exactly four pibs. Con... | {
"ground_truth": "all",
"style": "rule"
} | {
"index": "38673",
"split": "train"
} |
math | [
{
"content": "In a country, there are 110 cities. Between each pair of cities, there is either a road or no road.\n\nA driver starts in a city with exactly one road leading out of it. After traveling along this road, he arrives at a second city, which has exactly two roads leading out of it. After traveling alo... | {
"ground_truth": "107",
"style": "rule"
} | {
"index": "38674",
"split": "train"
} |
math | [
{
"content": "Let \\( f : \\mathbb{C} \\to \\mathbb{C} \\) be defined by \\( f(z) = z^2 - 2iz + 2 \\). Determine the number of complex numbers \\( z \\) such that \\( \\text{Im}(z) > 0 \\) and both the real and the imaginary parts of \\( f(z) \\) are integers within the range \\(-5\\) to \\(5\\). Please output ... | {
"ground_truth": "143",
"style": "rule"
} | {
"index": "38675",
"split": "train"
} |
math | [
{
"content": "In a district of Shanghai, the government convened the heads of 5 companies for an annual experience exchange meeting. Among them, Company A had 2 representatives attending, while the other 4 companies each had 1 representative attending. If 3 representatives are to be selected to speak at the mee... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "38676",
"split": "train"
} |
math | [
{
"content": "Determine the residue of $-811\\pmod{24}$. Your answer should be an integer in the range $0,1,2,\\ldots,22,23$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "38677",
"split": "train"
} |
math | [
{
"content": "In a group photo, 4 boys and 3 girls are to stand in a row such that no two boys or two girls stand next to each other. How many different arrangements are possible? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "38678",
"split": "train"
} |
math | [
{
"content": "Someone observed that $6! = 8 \\cdot 9 \\cdot 10$. Find the largest positive integer $n^{}_{}$ for which $n^{}_{}!$ can be expressed as the product of $n - 3_{}^{}$ consecutive positive integers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "38679",
"split": "train"
} |
math | [
{
"content": "The sum $10 e^{2 \\pi i/11} + 10 e^{15 \\pi i/22}$ is expressed as $re^{i \\theta}.$ Enter the ordered pair $(r, \\theta).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\left( 10 \\sqrt{2}, \\frac{19 \\pi}{44} \\right)",
"style": "rule"
} | {
"index": "38680",
"split": "train"
} |
math | [
{
"content": "Let $n \\geq 2$ be a natural. Define \n$$X = \\{ (a_1,a_2,\\cdots,a_n) | a_k \\in \\{0,1,2,\\cdots,k\\}, k = 1,2,\\cdots,n \\}$$.\nFor any two elements $s = (s_1,s_2,\\cdots,s_n) \\in X, t = (t_1,t_2,\\cdots,t_n) \\in X$, define \n$$s \\vee t = (\\max \\{s_1,t_1\\},\\max \\{s_2,t_2\\}, \\cdots , \... | {
"ground_truth": "(n + 1)! - (n - 1)!",
"style": "rule"
} | {
"index": "38681",
"split": "train"
} |
math | [
{
"content": "What is the value of $\\sqrt{5! \\cdot (5!)^2}$ expressed as a positive integer? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "240\\sqrt{30}",
"style": "rule"
} | {
"index": "38682",
"split": "train"
} |
math | [
{
"content": "Determine for how many different values of $p<2000$, there exists a quadrilateral $ABCD$ with integer side lengths, perimeter $p$, right angles at $D$ and $C$, $AB=3$, and $CD=AD$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "996",
"style": "rule"
} | {
"index": "38683",
"split": "train"
} |
math | [
{
"content": "If $x^2 + 2x + 5$ is a factor of $x^4 + Px^2 + Q,$ find $P + Q.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "38684",
"split": "train"
} |
math | [
{
"content": "A water tower in the shape of a cylinder has radius 10 m and height 30 m. A spiral staircase, with constant slope, circles once around the outside of the water tower. A vertical ladder of height 5 m then extends to the top of the tower. What is the total distance along the staircase and up the lad... | {
"ground_truth": "72.6 \\mathrm{~m}",
"style": "rule"
} | {
"index": "38685",
"split": "train"
} |
math | [
{
"content": "Find the positive value of $n$ such that the equation $9x^2+nx+1=0$ has exactly one solution in $x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "38686",
"split": "train"
} |
math | [
{
"content": "Calculate the value of the expression \\[(5^{1003}+6^{1004})^2-(5^{1003}-6^{1004})^2\\] and express it in the form of $k\\cdot30^{1003}$ for some positive integer $k$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "38687",
"split": "train"
} |
math | [
{
"content": "Given a complex number $z=3+bi$ ($b\\in\\mathbb{R}$), and $(1+3i) \\cdot z$ is a pure imaginary number. \n(1) Find the complex number $z$; \n(2) If $w= \\frac{z}{2+i}$, find the modulus of the complex number $w$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "38688",
"split": "train"
} |
math | [
{
"content": "Let $b_n$ be the number obtained by writing the integers $1$ to $n$ from left to right, and then reversing the sequence. For example, $b_4 = 43211234$ and $b_{12} = 121110987654321123456789101112$. For $1 \\le k \\le 100$, how many $b_k$ are divisible by 9? Please output the final answer within \\... | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "38689",
"split": "train"
} |
math | [
{
"content": "A parabola is given by the equation $y^{2}=4x$. Let $F$ be its focus, and let $A$ and $B$ be the points where a line passing through $F$ intersects the parabola. Let $O$ be the origin of the coordinate system. If $|AF|=3$, find the area of $\\triangle AOB$. Please output the final answer within \\... | {
"ground_truth": "\\frac{3\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "38690",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, it is known that $BC = 1$, $\\angle B = \\frac{\\pi}{3}$, and the area of $\\triangle ABC$ is $\\sqrt{3}$. The length of $AC$ is __________. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{13}",
"style": "rule"
} | {
"index": "38691",
"split": "train"
} |
math | [
{
"content": "From 5 male doctors and 4 female doctors, 3 doctors are to be selected to form a medical team, with the requirement that both male and female doctors must be included. Calculate the total number of different team formation plans. Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "70",
"style": "rule"
} | {
"index": "38692",
"split": "train"
} |
math | [
{
"content": "Julia's garden has a 3:7 ratio of tulips to daisies. She currently has 35 daisies. She plans to add 30 more daisies and wants to plant additional tulips to maintain the original ratio. How many tulips will she have after this addition? Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "38693",
"split": "train"
} |
math | [
{
"content": "The dimensions of a part on a drawing are $7{}_{-0.02}^{+0.05}$ (unit: $mm$), indicating that the maximum requirement for processing this part should not exceed ______, and the minimum should not be less than ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6.98",
"style": "rule"
} | {
"index": "38694",
"split": "train"
} |
math | [
{
"content": "One face of a pyramid with a square base and all edges of length 2 is glued to a face of a regular tetrahedron with edge length 2 to form a polyhedron. What is the total edge length of the polyhedron? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "38695",
"split": "train"
} |
math | [
{
"content": "Compute $\\tan \\left(\\frac{\\pi}{7}\\right) \\tan \\left(\\frac{2 \\pi}{7}\\right) \\tan \\left(\\frac{3 \\pi}{7}\\right)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{7}",
"style": "rule"
} | {
"index": "38696",
"split": "train"
} |
math | [
{
"content": "It is known that there is a four-person team consisting of boys A and B and girls C and D participating in a quiz activity organized by a TV station. The activity consists of four rounds. Let the probabilities for the boys to pass rounds one to four be $\\frac{5}{6}, \\frac{4}{5}, \\frac{3}{4}, \\... | {
"ground_truth": "\\frac{16}{15}",
"style": "rule"
} | {
"index": "38697",
"split": "train"
} |
math | [
{
"content": "A notebook sheet is painted in 23 colors, with each cell in the sheet painted in one of these colors. A pair of colors is called \"good\" if there exist two adjacent cells painted in these colors. What is the minimum number of good pairs? Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "38698",
"split": "train"
} |
math | [
{
"content": "A rigid board with a mass of \\(m\\) and a length of \\(l = 24\\) meters partially lies on the edge of a horizontal surface, hanging off by two-thirds of its length. To prevent the board from falling, a stone with a mass of \\(2m\\) is placed at its very edge. How far from the stone can a person w... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "38699",
"split": "train"
} |
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